Nonlinear Phase Unwinding of Functions

By Ronald R. Coifman, Stefan Steinerberger. 2015

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We study a natural nonlinear analogue of Fourier series. Iterative Blaschke factorization allows one to formally write any holomorphic function F as a series which successively unravels or unwinds the oscillation of the function $$\begin{aligned} F = a_1 B_1 + a_2 B_1 B_2 + a_3 B_1 B_2 B_3 + \cdots \end{aligned}$$F=a1B1+a2B1B2+a3B1B2B3+⋯where $$a_i \in \mathbb {C}$$ai∈C and $$B_i$$Bi is a Blaschke product. Numerical experiments point towards rapid convergence of the formal series but the actual mechanism by which this is happening has yet to be explained. We derive a family of inequalities and use them to prove convergence for a large number of function spaces: for example, we have convergence in $$L^2$$L2 for functions in the Dirichlet space $$\mathcal {D}$$D. Furthermore, we present a numerically efficient way to expand a function without explicit calculations of the Blaschke zeroes going back to Guido and Mary Weiss.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/cfd71636b19a9780a94add1028cdb58e5ea4efa5